Chin. Phys. B, 2021, Vol. 30(6): 060202    DOI: 10.1088/1674-1056/abd7e3
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# Soliton, breather, and rogue wave solutions for solving the nonlinear Schrödinger equation using a deep learning method with physical constraints

Jun-Cai Pu(蒲俊才)1, Jun Li(李军)2, and Yong Chen(陈勇)1,3,4,†
1 School of Mathematical Sciences, Shanghai Key Laboratory of Pure Mathematics and Mathematical Practice, and Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200241, China;
2 Shanghai Key Laboratory of Trustworthy Computing, East China Normal University, Shanghai 200062, China;
3 College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China;
4 Department of Physics, Zhejiang Normal University, Jinhua 321004, China
Abstract  The nonlinear Schrödinger equation is a classical integrable equation which contains plenty of significant properties and occurs in many physical areas. However, due to the difficulty of solving this equation, in particular in high dimensions, lots of methods are proposed to effectively obtain different kinds of solutions, such as neural networks among others. Recently, a method where some underlying physical laws are embeded into a conventional neural network is proposed to uncover the equation's dynamical behaviors from spatiotemporal data directly. Compared with traditional neural networks, this method can obtain remarkably accurate solution with extraordinarily less data. Meanwhile, this method also provides a better physical explanation and generalization. In this paper, based on the above method, we present an improved deep learning method to recover the soliton solutions, breather solution, and rogue wave solutions of the nonlinear Schrödinger equation. In particular, the dynamical behaviors and error analysis about the one-order and two-order rogue waves of nonlinear integrable equations are revealed by the deep neural network with physical constraints for the first time. Moreover, the effects of different numbers of initial points sampled, collocation points sampled, network layers, neurons per hidden layer on the one-order rogue wave dynamics of this equation have been considered with the help of the control variable way under the same initial and boundary conditions. Numerical experiments show that the dynamical behaviors of soliton solutions, breather solution, and rogue wave solutions of the integrable nonlinear Schrödinger equation can be well reconstructed by utilizing this physically-constrained deep learning method.
Keywords:  deep learning method      neural network      soliton solutions      breather solution      rogue wave solutions
Received:  17 December 2020      Revised:  29 December 2020      Accepted manuscript online:  04 January 2021
 PACS: 02.30.Ik (Integrable systems) 05.45.Yv (Solitons) 07.05.Mh (Neural networks, fuzzy logic, artificial intelligence)
Fund: Project supported by the National Natural Science Foundation of China (Grant No. 11675054), the Fund from Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213), and the Project of Science and Technology Commission of Shanghai Municipality (Grant No. 18dz2271000).
Corresponding Authors:  Yong Chen     E-mail:  ychen@sei.ecnu.edu.cn